"All Star" by Smash Mouth is not only considered the Signature Song of the entire Shrek franchise but a great one because it captures his humorous and great nature. "Somebody once told me the world is gonna roll me..."
"Bad Reputation" captures the spirit of Shrek defeating all of Lord Farquaad's knights nicely because of the rocking beats.
"I'm On My Way" by The Proclaimers is incredibly catchy and upbeat for the scene where Shrek and Donkey set out on their journey to rescue Fiona, and the Scottish accents fit all too well, to the point where you could imagine it taking place from Shrek's perspective.
John Cale's cover of "Hallelujah" is fondly remembered for its beautiful and graceful way of swaying the audience.
The franchise's main theme, Fairytale, a hauntingly beautiful leitmotif that still manages to fit with the series' tongue-in-cheek tone. Added bonus for that basically becoming the unofficial opening song of all the Dreamworks films to come after - whenever they do their Dreamworks Logo Gag, expect to hear some variation of it.
Both covers of "I'm a Believer" (Eddie Murphy's in the movie, Smash Mouth's on the soundtrack) are awesome, but Donkey belting it out at the end of the first movie with a crowd of fairy tales backing him up deserves special mention. For a typical kid watching this movie back in the summer of 2001, that whole sequence was the epitome of cool.
Aside from the overly high note, Fiona singing with the bird is nice to hear, especially with the music that plays with it. It almost sounds like something out of Disney and makes you wish the song wasn't cut so short.